Hi Ralph!
Your posts have been very helpful overall, but the problem still remains. I have a client with a Mac Pro running 10.9.5. He was previously able to screen share with no issues. As of about 2-3 weeks ago, he no longer has this capability. The invitation is accepted and the text displays on the screen announcing the session has begun, audio chat is possible, but the screen sharing never starts in earnest, and after a time there is an error for lack of response.
Valid AIM-to-AIM or Jabber (both tried)
No firewall in privacy and security (have toggled on and off and made sure Messages was allowed)
Screen sharing enabled in System Preferences
Caches dumped, iChat container emptied
Permissions repaired
In-place reinstall 10.9.5
Permissions repaired again
Disk repaired
New account created
Second user account created
When Screen Sharing app is accessed directly, it functions across the network.
Same result for all
Access is through Time Warner Cable Business class. Motorola SB6850 modem/router
End user cannot access router settings, so there may be some mysteries in there. The DNS servers are being read from the router and can be seen in Network>Advanced>TCP/IP (Should these be copied in directly, or is the fact that they are visible enough?)
Wirelessly or wired, same result.
At this stage, it is looking like a clean installation of either Mavericks or perhaps Yosemite may be order, unless you can pull a rabbit out of your hat.
Thanks for your time, effort and expertise!